Timeline for "Spatial (geometrical)" realization of Elementary topos?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 26, 2016 at 20:42 | comment | added | Qiaochu Yuan | @Simon: yes, I required the tensor product to distribute over colimits above. Elementary topoi can be ind-completed and I think the result is now monoidal cocomplete although I haven't checked this. | |
| Apr 26, 2016 at 20:38 | comment | added | Simon Henry | By monoidal cocomplete I think you mean also closed. or at least such that the tensor product commutes to co-limit in each variable (which is the same as closed if everything is presentable). But I don't see the relation with elementary toposes... | |
| Apr 26, 2016 at 20:06 | comment | added | Andrej Bauer | In that case, I would liken finitely cocomplete categories to abelian groups. | |
| Apr 26, 2016 at 20:01 | comment | added | Qiaochu Yuan | @Andrej: well, that's stronger than an analogy, right? One is even a special case of the other. I really just mean an analogy here. It's very naive: colimits are like addition. A more precise analogy would have been to commutative monoids since coproducts don't have inverses. For example, like in commutative monoids, there is a zero object, and biproducts. In this analogy presheaf categories are analogous to free abelian groups. | |
| Apr 26, 2016 at 19:43 | comment | added | Andrej Bauer | Complete categories are analogous to Abelian groups? Can you explain a bit more? I could say "cocomplete categories are analogous to sup-complete lattices", for instance, and that makes sense to me. I am curious how you get to Abelian groups. | |
| Apr 26, 2016 at 18:32 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
added 89 characters in body
|
| Apr 26, 2016 at 18:19 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |